### A characterization of ${B}^{*}$-algebras.

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A constructive proof of the Beurling-Rudin theorem on the characterization of the closed ideals in the disk algebra A(𝔻) is given.

Consider, by way of example, the following F. and M. Riesz theorem for Rn: Let μ be a finite measure on Rn whose Fourier transform μ* is supported in a closed convex cone which is proper, that is, which contains no entire line. Then μ is absolutely continuous (cf. Stein and Weiss [SW]). Here, as in the sequel, absolutely continuous means with respect to Lebesque measure. In this theorem one can replace the condition on the support of μ* by a similar condition on the wave front set WF(μ) of μ, while...

We present a change of variable method and use it to prove the equivalence to bundle shifts for certain analytic Toeplitz operators on the Banach spaces ${H}^{p}\left(G\right)(1\le p<\infty )$. In Section 2 we see this approach applied in the analysis of essential spectra. Some partial results were obtained in [9] in the Hilbert space case.

Let H²(bΩ) be the Hardy space of a bounded weakly pseudoconvex domain in ${\u2102}^{n}$. The natural resolution of this space, provided by the tangential Cauchy-Riemann complex, is used to show that H²(bΩ) has the important localization property known as Bishop’s property (β). The paper is accompanied by some applications, previously known only for Bergman spaces.

We show that a theorem of Rudin, concerning the sum of closed subspaces in a Banach space, has a converse. By means of an example we show that the result is in the nature of best possible.